[#P41] Union-closed sets conjecture

Problem. If \(\mathcal{F}\) is a finite nonempty family of finite sets satisfying \(A\cup B\in\mathcal{F}\) for all \(A,B\in\mathcal{F}\), then some element belongs to at least \(|\mathcal{F}|/2\) members of \(\mathcal{F}\).

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Definitions and notation

1Context

The conjecture, also called Frankl's conjecture, asks whether closure under union forces one element to appear with frequency at least one half.

2Definitions

Definition 1 (A family). A family is union-closed when the union of any two member sets is also a member.

Definition 2 (The frequency of an element). The frequency of an element is the number of member sets containing it.

3What counts as a solution

Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses. Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample.^[1][2][3]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses. Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses. Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample.^[1][2][3] Relevance to this problem For Union-closed sets conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses. Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample. Open full record →
Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets.^[1] Relevance to this problem For Union-closed sets conjecture, record union-closed-sets-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2024 paper identifies the union-closed sets conjecture as open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Strong asymptotic lower bounds and many finite or structural cases are known, while the one-half threshold remains open.

Recorded example 1. For the family {empty set, {1}, {1,2}}, the element 1 appears in two of the three sets.

Computational notes

Exhaustive family generation can verify bounded universe or family sizes only.

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“Union-closed sets conjecture.” TheoremDB. P41. Problem statement; statement text SHA-256 03f9357bf91fd8a74ff8c8123e3c4002a7df5d36a062cffbba3783d357e23b5d. https://theoremdb.org/statement/?ref=P41

BibTeX

@misc{theoremdb-problem-03f9357bf91fd8a74ff8c8123e3c4002a7df5d36a062cffbba3783d357e23b5d,
  title = {{Union-closed sets conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 03f9357bf91fd8a74ff8c8123e3c4002a7df5d36a062cffbba3783d357e23b5d},
  url = {https://theoremdb.org/statement/?ref=P41}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Kengbo Lu and Abigail Raz, “Note on the union-closed sets conjecture and Reimer's average set size theorem”. arXiv:2405.10639 (2024). arXiv:2405.10639, abstract and introduction. ↗preprint · primary source · arXiv:2405.10639, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2024 paper identifies the union-closed sets conjecture as open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and introduction.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked open-status source.Source named by the research packet.
Justin Gilmer, “A constant lower bound for the union-closed sets conjecture”. arXiv:2211.09055 (2022). Abstract. ↗preprint · primary source · arXiv:2211.09055v2 · checked 2026-08-01Source use: original summary.First universal constant lower bound.
Lei Yu, “Dimension-Free Bounds for the Union-Closed Sets Conjecture”. Entropy 25(5) (2023), 767. DOI 10.3390/e25050767. Abstract. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Computable entropy bound evaluated at approximately 0.38234.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P41] Union-closed sets conjecture

1Context

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3What counts as a solution

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Computational notes

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